0:10Skip to 0 minutes and 10 secondsComplex systems can often be depicted or modelled as networks. The mathemetical tool for network analysis is Graph Theory. The basic tools are nodes and edges. In this lecture, I will introduce you to two types of networks-- random networks and scale-free networks. Focus on the nodes and the number of edges that they have. Count the number of edges a node has. Put them in a picture. There's some left-to-right nodes with an increasing number of edges, from zero edges on the left to the highest number of edges found in a network on the right. What shows up as a distribution? In this case, showing the number of edges or nodes in a network. Other characteristics can also be plotted in a distribution.

1:18Skip to 1 minute and 18 secondsIn network theory, so-called random networks are defined. In a random network, the probability of a node being connected to any other node is identical. And this leads to a normal distribution of the number of connections, a bell curve or a Gauss curve after the mathematician who has invented this distribution. The normal distribution is very common, often found in the real world, but interestingly enough, not in complex networks. See also the first lecture. The Pareto or power law distributions are very common. They are different from the normal distribution. Why do random networks not occur very often? Networks grow over time. They are not created or designed overnight. They evolve. The Hypothesis of Preferential Attachment leads to scale free networks.

2:22Skip to 2 minutes and 22 secondsThey have fatter tails, extreme events occur more frequently. Highly connected nodes are more common. There exist more of them than the normal distribution, which suggests this is very relevant among other things, for policy and risk management. You underestimate occurrence of low-probability, high-impact events. Black swans. If you assume a normal distribution ready actual distribution has a fatter tail.

2:54Skip to 2 minutes and 54 secondsPower law distributions are also found in the distribution of cities by size, so-called Zipf's Law in income and wealth distributions. Your financial market prices, for example, stock prices. Then we come back to the Theory of Preferential Attachments. What the theory says is that there is a higher probability of attaching to nodes that's already have a higher number of edges. You can understand why that is the case, it's more attractive to connect to somebody who is already connected to many other people. And that then, in the end, leads to certain classes of power law distributions. The Theory of Preferential Attachment can also be used to explain the distribution of other characteristics. There are also other explanations for generating scale free networks.

3:45Skip to 3 minutes and 45 secondsThis is still an area up for research. In this lecture, we have learned what random and scale free networks are. And we have discussed one reason why scale free networks are far more common than random networks in reality.

Random & scale-free networks

This lecture will introduce the concepts of Random and Scale free networks.

The distinction that we make between networks comes from the distribution of edges. In other words, how connected are the different nodes and how different are these connections? Do all nodes have more or less the same amount of edges? Or do some nodes have a bigger chance of connecting to other nodes?